Quadrature solution for the general relativistic motion of a satellite or a planet

Abstract

The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle ϕ as a quasi-periodic function ϕ (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17″ of arc per year. The periodic part is a sine series, with coefficients containing the small parameter β≡2GM/c2p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off.